Operator Norm Bounds for Multi-leg Matrix Tensors and Applications to Random Matrix Theory

TL;DR

This paper develops a graphical formalism to bound partial traces of matrix tensors under operator norm constraints and applies these bounds to advance understanding in random matrix theory.

Contribution

It introduces a novel graphical framework for analyzing multi-leg partial traces and derives sharp bounds, extending asymptotic freeness concepts in random matrix models.

Findings

Established exact bounds for partial traces based on graph cycles.

Derived operator norm estimates for matrices from partial permutations.

Extended asymptotic freeness to matrix coefficient algebras in random matrix models.

Abstract

We investigate the extremal values of partial traces of matrix tensors under operator norm constraints. To evaluate these multi-linear quantities, we develop a comprehensive graphical formalism that encodes multi-leg partial traces, partial permutations, and their moments using colored directed graphs. With this graphical framework, we establish optimal, sharp bounds for the partial trace $(\mathrm{Tr}_{\sigma_1} \otimes \ldots \otimes \mathrm{Tr}_{\sigma_k})(A_1, \ldots, A_m)$ over matrices bounded by $\|A_i\| \le 1$. Specifically, we prove that this maximum evaluates exactly to $N^{M(\sigma_1,\ldots,\sigma_k)}$, where $N$ is the dimension and $M$ represents the maximal number of directed cycles in the associated graph across all possible internal vertex pairings. We further derive explicit operator norm estimates for matrices generated by partial traces of partial permutations. Finally, we apply these combinatorial bounds to multi-matrix random matrix theory. By examining models involving Ginibre ensembles, we extend concepts of asymptotic freeness to matrix coefficient algebras, establishing operator norm estimates that rigorously separate the asymptotic behavior of non-crossing and crossing pairings.